Truth-functional vs. Non-truth-functional Operators

 
 The Rules of Truth for PL allow us to compute the truth-value of any wff of PL, if we know the truth-value of its sentence letters.  The truth-value of any complex wff of PL, then is completely determined by the truth value of its component sentence letters.  We express this fact by saying that the operators of PL are truth-functional, and that the system of PL is truth-functional, since all its operators are truth-functional.

Truth-functionality is best understood by comparing truth-functional operators with operators from a natural language like English which are not truth functional.  Consider the following sentences:
 
 

Sentences 1 and 2 are simple sentences, which could be abbreviated with sentence letters.  Sentences 3 and 4 are complex, constructed out of sentences 1 and 2 respectively by prefixing the "Ignat believes that" operator.  Let's suppose that 1, 2 and 3, are assigned the truth-value "T".  If "Ignat believes that" were a truth-functional operator, what truth-value would we have to assign to sentence 4?

Sentence 3 takes "T" as its input - the truth-value of its part, and it returns "T" as the truth-value for the whole. Sentence 4 also has "T" as it's input. If 4 were truth-functional, its truth-value would also be completely determined by the truth-value of its part, which has the same truth-value as sentence 3, and so would have to be "T".  But it's obvious that 1,2 and 3 could be true, and 4 false. Ignat might not even read Snow Country!

There are two exercises on this distinction. The first asks you to explain why "and then" is not truth-functional, and the second asks you to explain why "Wayne hopes that" is not truth-functional. Here is an exemplary answer for the second. But look at it only after you've made your own attempt!